Chemistry Reference
In-Depth Information
x
2
L
2
x
2
Unique equilibrium
R
1
.x
2
/
x
2
R
2
.x
1
/
x
1
L
1
x
1
x
1
Fig. 1.2
Example 1.2; the Cournot model with linear price function and quadratic cost function in
the case of duopoly .N
2/. The reaction functions R
1
.x
2
/;R
2
.x
1
/ and the unique equilibrium.
The figure illustrates case (ii) when B
2
<4.B
D
e
2
/ and x
k
>x
k
, k
C
e
1
/.B
C
D
1;2
(ii) Assume next that for all k,
B<e
k
<0, then the cost function is concave,
however '
k
remains concave in x
k
, so the best response remains the same as
above. However, this case raises the possibility of multiple equilibria. Consider
a duopoly (N
D
2). Figure 1.2 depicts the reaction functions in the case where
B
2
<4.B
C
e
1
/.B
C
e
2
/;
that is when marginal costs are decreasing but not too strongly.
2
Furthermore,
the “limit quantities” x
k
D
.A
c
k
/=B, that is the corresponding quantity lev-
els which guarantee that the other firm is kept out of the market, are larger than
the monopoly quantities x
k
D
.A
c
k
/=.2.B
C
e
k
//. Under these conditions
there is still a unique interior equilibrium given by
E
D
.x
1
; x
2
/
2.B
C
e
2
/.A
c
1
/
B.A
c
2
/
4.B
C
e
1
/.B
C
e
2
/
B
2
D
;
2.B
C
e
1
/.A
c
2
/
B.A
c
1
/
4.B
C
e
1
/.B
C
e
2
/
B
2
(1.12)
2
This interpretation is based on the fact that the condition is satisfied if
e
k
.k
D
1;2/ does not
get too close to B.
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